(of a ship) be registered to a particular country and sail under its flag.

represent or demonstrate support for your country, political party or organization especially when you are abroad.

The forms show the flag, carry the flag and wave the flag are also found.

1996 – Hello! - She flew the flag for British tennis in the

fly high

be very successful

prosper

The noun high-flyer (or high-flier) meaning a “successful and ambitious person" developed from this phrase in the mid 17th century.

a fly in amber

a curious relic of the past preserved into the present

The image is of the fossilized bodies of insects which are often found preserved in amber.

fly in the face of

be openly at variance with what is usual or expected.

a fly in the ointment

a minor irritation or other factor that spoils the success or enjoyment of something

This expression alludes to Ecclesiastes 10 : 1 : Dead flies cause the ointment of the apothecary to send forth a stinking savour.

1998 - Times - Before you conclude that I have become a raging Europhile, let me say that there is a fly in the ointment.

fly a kite

try something out to test opinion – informal

A historical sense of this phrase was raise money by an accommodation bill meaning to raise money on credit and this sense of testing public opinion of your credit worthiness gave rise to the current figurative sense. The US phrase go fly a kite means go away.

fly the nest

(of a young person) leave their parent's home to set up home elsewhere – informal

The image here is of a young bird's departure from its nest on becoming able to fly.

fly off the handle

lose your temper suddenly and unexpectedly – informal

This expression uses the image of a loose head of an axe flying off its handle while the axe is being swung.

a fly on the wall

an unnoticed observer of a particular situation

This expression is often used as an adjective as in a fly-on-the-wall documentary where it refers to a film making technique in which events are merely observed and presented realistically with minimum interference, rather than acted out under direction.

a fly on the wheel

a person who overestimates their own influence

This phrase stems from Aesop's fable of a fly sitting on the axletree of a moving chariot and saying - See what a dust I raise.

like a blue-arsed fly

in an extremely hectic or frantic way - British vulgar slang

The blue-arsed fly referred to is a bluebottle, well known for its frenetic buzzing about.

1998 - Rebecca Ray - A Certain Age - I'm not going to run around like a blue-arsed fly pandering to you and your bloody room, alright?

on the fly

while in motion

while busy or active

(of an addition or modification in computing) carried out during the running of a program without interrupting the run

there are no flies on him

the person mentioned is very quick and astute

Early instances of this expression suggest that it originated with reference to cattle who were so active that no flies settled on them. The phrase was noted in the mid 19th century as being very common in Australia as a general expression of approbation. In the USA it could also be used to convey that the person in question was of superior breeding or behaved honestly.